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Find the derivative of y = e^sin x....

Find the derivative of `y = e^sin x`.

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To find the derivative of the function \( y = e^{\sin x} \), we can follow these steps: ### Step 1: Identify the function The given function is: \[ y = e^{\sin x} \] ### Step 2: Use the chain rule To differentiate \( y \) with respect to \( x \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative is given by: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] where \( u = g(x) \). ### Step 3: Set \( u \) Let: \[ u = \sin x \] Then, we can rewrite the function as: \[ y = e^u \] ### Step 4: Differentiate \( y \) with respect to \( u \) Now, differentiate \( y \) with respect to \( u \): \[ \frac{dy}{du} = e^u \] ### Step 5: Differentiate \( u \) with respect to \( x \) Next, differentiate \( u \) with respect to \( x \): \[ \frac{du}{dx} = \cos x \] ### Step 6: Apply the chain rule Now, substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot \cos x \] ### Step 7: Substitute back for \( u \) Finally, substitute \( u = \sin x \) back into the equation: \[ \frac{dy}{dx} = e^{\sin x} \cdot \cos x \] ### Final Answer Thus, the derivative of \( y = e^{\sin x} \) is: \[ \frac{dy}{dx} = \cos x \cdot e^{\sin x} \] ---
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